Multiples
Category : 5th Class
Multiples
When two or more than two numbers are multiplied with each other, the resulting number is the multiple of all that numbers. For example, if A x B = C, C is multiple of both A and B.
Multiples of 5 = 5, 10, 15, 20, 25, ____
Multiples of 2 = 2, 4, 6, 8, 10, 12,____
Multiples of 10 = 10, 20, 30, 40, ____
Common Multiples
The same multiples of two or more than two different numbers are called common multiples.
Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40,__
Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45,___
Common multiples of 4 and 5 = 20, 40, 60, etc.
Least Common Multiple (L.C.M.) The least common multiple among the common multiples of two or more than two numbers is called least common multiple.
Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ___
Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ___
Common multiples of 5 and 7 = 35, 70,___
Least common multiple of 5 and 7 = 35.
LCM by Prime Factorization Method
In the prime factorization method numbers are written in the form of product of prime factors. The common and non-common factors are separated out and multiplied with each other. Their product are the LCM of the given numbers.
Prime factorization of \[28=2\times 2\times 7\]
Prime factorization of \[36=2\times 2\times 3\times 3\]
Common prime factors \[=2\times 2\]
Non-common factors \[=7\times 3\times 3\]
Product of the common factors \[=2\times 2\times 3\times 3\times 7=252\]
Thus LCM of 28 and 36 = 252.
LCM by Division Method
Division method of LCM involves the following steps:
Step 1: Write the numbers whose LCM is to be found in a row.
Step 2: Choose the least prime number by which at least two numbers are divisible. Then divide the numbers by the prime number.
Step 3: Write the quotient and undivided number in the next row just below the respected number.
Step 4: Repeat the process of division by the prime numbers unless only 1 or co-prime numbers remain in the last row.
Step 5: Now multiply the divisor and co-prime numbers of the last row. Product is the LCM.
Find the LCM of 12, 20, 32 and 28.
| 2 | 12, 20, 32, 28 |
| 2 | 6, 10, 16, 14 |
| 3 , 5, 8, 7 |
\[2\times 2\times 3\times 5\times 8\times 7=3360.\]
Relation between Two Numbers and Their LCM and HCF
(a) \[LCM\times H.C.F=\]Product of the two numbers
(b)\[\text{LCM=}\frac{\text{Product of the numbers}}{\text{H}\text{.C}\text{.F}}\]
(c) \[\text{HCF=}\frac{\text{Product of the numbers}}{\text{L}\text{.C}\text{.M}}\]
(d) Required number\[\text{=}\frac{\text{L}\text{.C}\text{.M }\!\!\times\!\!\text{ H}\text{.C}\text{.F}}{\text{Given}\,\text{number}}\]
Find the LCM and HCF of 40 and 45 and verify that product of 40 and 45 is equal to the product of their LCM and HCF.
LCM of 40 and 45 =360
HCF of 40 and 45 =5
Product of the numbers \[=40\times 45=1800\]
Product of their HCF and \[\text{LCM}=360\times 5=1800.\]
If HCF and LCM of the two numbers are 10 and 300 respectively and one of the numbers is 50 find the other number.
Solution:
Required number\[\text{=}\frac{\text{L}\text{.C}\text{.M }\!\!\times\!\!\text{ H}\text{.C}\text{.F}}{\text{Given}\,\text{number}}\] \[\text{=}\frac{\text{10}\,\text{ }\!\!\times\!\!\text{ }\,\text{300}}{50}\] = 60.
Which one of following sets contains all the factors of 72?
(a) {1,2, 4, 8, 9, 6, 12, 36, 72}
(b) {1,2, 3, 4, 6, 8, 9, 12, 18, 24, 48, 72}
(c) {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}
(d) {2, 3, 4, 6, 8, 9, 12,18, 24, 36, 48, 72}
(d) None of these
Answer: (c)
Explanation
Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36,and 72
When a number is divided by its factor the remainder will be__.
(a) 1
(b) 10
(c) 0
(d) 9
Answer: (c)
Explanation
A number is completely divisible by its factors leaving no remainder.
Which one of the following digits should be placed in the middle of the digits of the number 258970 so that 3 becomes factor of it?
(a) 0
(b) 1
(c) 2
(d) 3
(e) None of these
Answer: (c)
Explanation
Sum of digits of the number \[258970=2+5+8+9+7+0=31.\]therefore, 2 should be placed in the middle of the digits such that sum of the digits becomes divisible by 3.
You have to add a least number in 1025629 in order to 12 be a factor of it. Choose the number from the following options and add.
(a) 10
(b) 11
(c) 12
(d) 8
(e) None of these
Answer: (b)
Which one of the following statements is not true about the factor?
(a) The greatest factor of a number is the number itself
(b) The smallest even number is a factor of all the even numbers
(c) If X and Y are the factors of each other then X = Y
(d) X and Y are two natural numbers. A is a factor of X and B is a factor of Y If X > Y Then A > B
(e) None of these
Answer: (d)
17, 19 is a pair of twin prime. To make it a prime triplet which one of the following prime numbers would you like to choose?
(a)11
(b) 13
(c) 17
(d) 19
(e) None of these
Answer: (b)
Explanation
17-13 = 4 ,
19 -13 = 6. Thus 13 differs from one of the numbers of twin prime by 4 and 6 from other;
If X is a composite number then what cannot be correct about X?
(a) Y is a prime number and a factor of X
(b) X is divisible by 2
(c) Sum of the smallest and the greatest factors of X is 1 more than X
(d) If X lies between 10 and 20, it has no prime factor
(e) None of these
Answer: (d)
Explanation
12,14,15,16,18, are the composite numbers which lie between 10 and 20. 12, 14, 16, 18 are divisible by 2 which is a prime number. Thus X has the prime factors even if it lies between 1 and 20.
Which one of the following is a perfect number?
(a) 8
(b) 10
(c) 14
(d) 28
(e) None of these
Answer: (d)
Q is a prime number and a member of a prime triplet. If Q = 17, which one of the following can not be a member of the prime triplet which contains Q?
(a) 11
(b) 13
(c) 19
(d) 29
(e) 23
Answer: (d)
Which one of the following is correct explanation for a prime number not to be a perfect number?
(a) Prime numbers and perfect numbers are two different numbers therefore, a prime number cannot be perfect number
(b) Only a composite number can be a perfect number because a prime number has only two factors
(c) A prime number cannot be a perfect number because it has only two factors 1 and the number itself and sum of these two factors cannot be twice of the number
(d) If sum of the factors of a number except the number is equal to the number then it is a perfect number
(e) None of these
Answer: (c)
Find the HCF of 256 and 400.
(a) 4
(b) 8
(c) 16
(d) 32
(e) None of these
Answer:(c)
Jack: HCF of two numbers remains same even if the numbers are multiplied by any number.
Codi: HCF of two numbers gets doubled when the numbers are doubled. Who is correct?
(a) Jack
(b) Codi
(c) Both are correct
(d) Both partially incorrect
(e) None of these
Answer: (b)
Explanation
Codi is correct because when two numbers are multiplied by a same number then their HCF also increase as the product of the original HCF and the number
Which one of the following statements is not true?
(a) If X and Y are the two co-prime numbers, their HCF is 1
(b) If HCF of X and Y is X then Y is divisible by X.
(c) If HCF of X and Y is equal to the HCF of Y and Z, HCF of X and Z is also the same
(d) lf X + l = Y, X and Y are co-prime numbers
(e) None of these
Answer: (c)
If Y = 2X, Z = 5Y, HCF of XYZ =:
(a) X
(b) Y
(c) Z
(d) All of these
(e) None of these
Answer: (a)
Sam chooses two numbers X and Y and he performs the long division method by dividing Y by X. He notices the last divisor is 1. Which one of the following is not true?
(a) X and Y have only one common factor
(b) X and Y are the co-prime numbers
(c) HCF of X and Y is 1
(d) X and Y are the even numbers
(e) None of these
Answer: (d)
Which one of the following is the LCM of 30 and 45?
(a) 45
(b) 90
(c) 180
(d) 360
(e) None of these
Answer: (b)
X is a 5 digit prime number. Sum of the digits of X is 28. You have to make it a multiple of least odd prime number by putting one more digit in the right side of the number. Which one of the following digits would you like to put?
(a) 0
(b) 1
(c) 2
(d) 3
(e) None of these
Answer: (c)
Explanation
Sum of the digits of X is 28 and the least odd prime number is 3.Jhus you should put 2 in the right side of the number so that sum of digits of the number be divisible by 3.
What two digit least number should be added to 25427 so that it becomes multiple of 25?
(a) 23
(b) 21
(c) 25
(d) 27
(e) None of these
Answer: (a)
X is a multiple of Y, Y is a multiple of Z, and Z is a multiple A and x > y. Which one of the following is the least common multiple of X, Y, Z, A?
(a) X
(b) Y
(c) Z
(d) A
(e) None of these
Answer: (a)
If LCM of two numbers is 600 and the numbers are 150 and 200 then find their HCF.
(a) 50
(b) 60
(c) 75
(d) 80
(e) None of these
Answer: (a)
Explanation
\[\text{HCF=}\frac{\text{Product of Numbers }\!\!~\!\!\text{ }}{L.C.M}=\frac{150\times 200}{600}=50\]
So the HCF = 50.
HCF and LCM of two numbers are 4 and 48 respectively. If one of the numbers is 2 more than X (Where X = 10) then find the other number.
(a) 10
(b) 12
(c) 16
(d) 18
(e) None of these
Answer: (c)
Explanation
Given X = 10 Therefore, one of the numbers = 10 + 2 = 12 Required number \[=\frac{4\times 48}{12}=16.\]
Jack: If X is a LCM of two co-prime numbers, X is equal to the product of the prime numbers.
Codi: HCF of two or more than two numbers is always a factor of their LCM. Who is correct?
(a) Jack
(b) Codi
(c) Both are correct
(d) Both partially incorrect
(e) None of these
Answer: (c)
Which one of the following statements is not correct?
(a) If X is a natural number and Y is a multiple of X then Y is the LCM of X and Y
(b) lf X = Y, LCM of X and Y = HCF of X and Y
(c) A number is the smallest multiple and greatest factor of itself
(d) LCM of two prime numbers is a prime number
(d) None of these
Answer: (d)
Find the least number when divided by 20, 25 and 50 leaves remainder 9 in each case?
(a) 89
(b) 109
(c) 209
(d) 229
(e) None of these
Answer: (b)
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